Philosophy is written in this grand book  I mean the universe
 which stands continually open to our gaze, but it cannot be
understood unless one first learns to comprehend the language and
interpret the characters in which it is written. It is written in
the language of mathematics, and its characters are triangles,
circles, and other geometrical figures, without which it is
humanly impossible to understand a single word of it; without
these, one is wandering about in a dark labyrinth.

Galileo Galilei (1564-1642)

I had been fascinated by the -2/3 slope since 1971 when I first
observed it, or one statistically indistinguishable from it, in the
"world regression line" (Chapter 5).

Between 1972 and 1975 a number of studies produced size-density slopes
which did not depart significantly from this -2/3 value. Each time it
happened I wondered Why? I tried to derive the slope theoretically
(i.e., mathematically, from initial assumptions), but I failed each time
until Spring term, 1975, when I was working with Jay Callen.

Dimensional Analysis: Time

Jay had been doing a lot of work
with "central place theory", much of which involves explaining empirical
regularities in terms of least distance models.[1] As it happened, I had been re-reading
Zipf's principal work in which empirical regularities were derived in
accordance with his principle of least effort.[2] At some point in all this I recalled a
remark made by Professor Anderson, in a graduate school seminar in
mathematical modeling, to the effect that well-developed theoretical
fields tend to select fundamental dimensions and define other
quantities in terms of these.

The classical example of such "dimensional analysis" is Newtonian Physics. Length is clearly a dimension, in a real sense the most basic one we have. With
length, plus a few other dimensions defined in terms of length (e.g., time
and mass  the distance a marker moves on a proper scale), other
quantities can be defined as in the following table, the last column of
which contains only s, m & t  length, mass and time:

Among other advantages, such reduction of all variables to one (or a few)
dimensions makes it possible to rigorously "balance equations"
dimensionally  have the same units on both sides of the equation, or
specify the units for embedded "constants." Different sciences use
different fundamental dimensions, e.g., Chemistry - energy; Economics -
money (but see the "Labor Theory of Value" below). I recall Professor Anderson
suggesting at some point in our conversations that one day, perhaps,
time might prove to be a fundamental dimension in Sociology. I
began to think about least-time.

Economics seems limited, as a general social science: what would
its least-cost model mean in primitive societies which lack a
money economy? In a similar vein, what could Geography's
least-distance theories contribute to an understanding of future
societies with "beam me up" technologies? I realize these are fictional,
but our transportation systems make our ancestors view of distance
hopelessly out of date also. Primtive societies where money is
meaningless, future societies where distance is meaningless  how do
you theorize about both using the same social science model? Zipf's
least-effort notion was a conscious attempt to combine monetary and
distance (and other) costs, but it smacked of mixing apples and oranges
(dollars + miles + sweat?), and it lacked rigor: how, exactly, do you
measure (quantify) "effort"?

The solution, it seems to me, is to translate these disparate variables
into the single dimension of Time. If I know the average velocity
of a given transportation technology, it is easy enough to translate
distance into time (distance is time, multiplied by veloctiy; time = distance/velocity). Depending upon
whether I use a horse or an automobile, the county seat may be either a
day or a

The Labor Theory of ValueWe
see then that that which determines the magnitude of the value of any
article is the amount of labour socially necessary, or the labour-time
socially necessary for its production. Each individual commodity, in this
connexion, is to be considered as an average sample of its class.
Commodities, therefore, in which equal quantities of labour are embodied,
or which can be produced in the same time, have the same value. The value
of one commodity is to the value of any other, as the labour-time
necessary for the production of the one is to that necessary for the
production of the other. "As values, all commodities are only definite
masses of congealed labour-time."

half-hour away. Time is money, according to Franklin; money is time, multiplied by the rate at which it accumulates (time =
money/rate): so much income, expenditure, rent, or interest per
hour, day, month or year. Even effort, though poorly defined itself, might
be translated into some expenditure of work-hours, given a level of
technology (e.g., how long would it take someone to dig a given trench
with no tools, with a shovel, with a backhoe?)

Pete PhillipsSan Francisco, June
1999

These thoughts took me back further in my education, to an undergraduate
course in the history of social thought, taught by Andrew P. Phillips at
what was then San Francisco State College. Along with the ideas of Comte,
Durkheim and Weber, it was there that I encountered the social thought of
Karl Marx and, incidentally, his labor theory of value: the value of a
commodity is a direct function of the average socially necessary labor
time required for its production. When I first heard of this idea I
appreciated the fact that, unlike most other social science theories, it
aimed at being cross-cultural, pan-historical, i.e., general 
presumably applicable to all societies, even non-monetized or
hyper-technological ones.

At the time I learned of these ideas of Marx, I was taking a Social
Psychology course in which I had occasion to review ideas of a number of
philosophers about the "essential difference" between human and animal
life. The text and teacher made much of so-called "symbolic
interactionism", the doctrine which asserts that the fundamental
difference between humans and animals is our reliance on "meaningful
symbols" (vs. mere "signs" like barks and chirps). It didn't seem to me that this was much of an
improvement over the ancient metaphysical assertion that we differ by
virtue of our capacity for thought (in fact, Aristotle's notion of us as
the zoon politikon seems a little more sophisticated). That was
back before contemporary scientists began studying animal language.
Against this notion that thought is what distinguishes us from
other animals, or that "symbolic interactionism" should be the
(dimensional?) basis of Sociology, let me quote one of my favorite lines
from Shakespeare:

But thought's the slave of life, and life time's fool;
And time, that takes survey of all the world, must have a stop.[3]

Thought's the slave of life (all life), and life is time's fool  it could as
easily have been written by Marx, or any biologist for that matter. I
don't understand why Sociology (other than the classic intro text
by the Lenskis) doesn't begin with Biology as Auguste Comte originally
suggested. It would be a small step from there, in the study of human
or non-human social systems, to see how relevant time really is
as a fundamental dimension.

Just for fun, Table 9-2 contains a very small sample of Shakespeare's many
other references to time. I once shut down an online literature search
engine by trying to get it to list all his references to "time"; next time
I tried it "time" had been removed from searches, treated as too common
along with words like "the" and "is"):

TABLE 8-2. A FEW TIME REFERENCES FROM
SHAKESPEARE

Time travels in divers paces with divers
person. I'll tell you who Time ambles
withal, who Time trots withal, who Time
Gallops withal, and who he stands still withal.
As You Like It, III,ii,328

Spite of cormorant devouring Time.
Love's Labors Lost, I,i,4

Come what may,
Time and the hour runs through
the roughest day.
Macbeth, I,iii,146

If you can look into the seeds of time,
And say which grain will grow and which will
not.
Macbeth, I,iii,58

To-morrow, and to-morrow, and to-morrow,
Creeps in this petty pace from day to day,
To the last syllable of recorded time;
And all our yesterdays have lighted fools
The way to dusty death. Out, out brief candle!
Life's but a walking shadow, a poor player
That struts and frets his hour upon the stage,
And then is heard no more: it is a tale
Told by an idiot, full of sound and fury,
Signifiying nothing.
Macbeth, V,v,17

A forted residence 'gainst the tooth of time,
And razure of oblivion.
Measure for Measure, V,i,12

There are many events in
the womb of time which will
be delivered.
Othello, I,ii,377

Time's the king of men.
He's both their parent, and he is their grave.
And gives them what he will, not what they
crave.
Pericles, II,iii,45

Like as the waves make towards the pebbled
shore
So do our minutes hasten to their end.
Sonnets, LX

What seest thou else
In the dark backward and abysm of time?
The Tempest, I,ii,49

Is there no respect of place, persons, nor time
in you?
Twelfth Night, II,iii,100

Time is like a fashionable host
That slightly shakes his parting guest by the
hand,
And with his arms outstretch'd as he would
fly,
Grasps in the comer ....
Troilus and Cressida, III,iii,165

Envious and calumniating time.
Troilus and Cressida, III,iii,174

The end crowns all,
And that old common arbitrator, Time,
Will one day end it.
Troilus and Cressida, IV,v,223

Time Minimization Theory

Begin with this: Assume that social structures evolve in such a way
as to minimize the time expended in their operation. Under this
constraint, determine how large a county (or other territorial division)
should be (in order to enforce law, maintain order, provide services, or
whatever).

One way of thinking about the value assumed by any variable is to assume
it to be the result of contradictory forces: those tending to increase it,
and those tending to decrease it. The larger a county is, other things
equal, the more people there are to pay for its operations. On the other
hand, the larger it is the harder it will be for distant citizens to reach,
or be reached by, the county seat of government.
Originally, the "maintenance" time of a county was donated by people who
volunteered their time as its officials. With the growth of county
government, however, the operations of a county are carried out by paid
employees. The money required to pay them comes from taxes and other
sources. Ultimately, however, we may imagine the money as the result of
time expenditures by those making up the county (e.g., as taxed income,
itself a product of time).

Sharing the maintenance costs creates a centrifugal force, a
tendency to increase the size the county, but as distance to the center
increases the "interaction" time also increases, creating a centripetal
force, a tendency to decrease county size. It seems reasonable to
assume that actual county sizes represent a balance of these opposing
tendencies, a minimizing of both "maintenance" and "interaction" times
somehow taken together.

We don't know what the maintenance time actually is, only that it
represents some expenditure of hours, h. We also don't know how the
burden is distributed among the citizens. We can refer to counties as
being more or less costly in terms of average or per-capita time
expenditures,

where P is the population of the county. By a similar argument,
ignoring the geographic distribution of the citizens within a county, we
can specify an average interaction time

where S is the average distance[4]
to the county seat and v is the average velocity under existing
transportation technology.

The time expended in the operation of this social structure is the
sum

We are trying to find an optimum value for A, the area served by
the county, but A does not appear in Eq. 2, at least not
directly. The definition of density is population divided by area

The average distance to the county seat must be proportional to the
square root of county area (by dimensional analysis a length in, e.g.,
miles must be proportional to the square root of an area in square
miles), so with the constant of proportionality, w, we obtain the
equation

With these substitutions we can re-write Eq. 2

(3)

We can now write T as a function of A, treating D, h, w and v as
(temporary) constants,

We want to express Area as a function of Density, so we extract it from the right-hand term of Eq. 6 to obtain

(7)

Reducing the constants to a single factor k

(8)

The logarithmic transformation of Eq. 8, with K = log k, is

(9)

Eq. 9 is identical to Eq. 1, the "world
regression line" which we sought to derive theoretically. Thus, it follows
directly from the assumption of time-minimization that the
size-density slope should be negative
two-thirds.

Lack of Intentionality in Time Minimization

I want to emphasize something about time-minimization as I
have used the term: I make no assumption about people consciously or
purposely minimizing time. Least-distance and
least-cost theories are often put forward as criteria for
rational decision making. Zipf presents his own theory of
least-effort as similarly rational, conscious, purposive. We feel
comfortable enough with models which describe the way a decision maker
makes rational decisions employing one or more of these criteria. And
rational decision makers certainly do try to manage time expenditure
(think of all the time-and-motion studies done in industry
throughout this century).

In contrast, I am explicitly not trying to specify rules to be
used by a rational decision maker. Nor am I in any way asserting that
people are inclined toward "rational" behavior. I am only trying to
account for the relationship described by Eq. 1. The
relationship between size and density results from, and describes, a
pattern of territorial division which is the product of countless numbers
of decisions, more-or-less rational, by countless numbers of
people, over very long periods of time and often huge expanses of space.
The actors involved were effectively unconscious of one another and
certainly unaware of the size-density relationship itself. I
have no reason to assume any of them consciously tried to establish a
negative two-thirdssize-density relation or
that they purposefully minimized time.

What I am putting forth here is pure theory, invention, fiction,
imagination. I am only trying to state the simplest assumption
which, if true, would account for the observed results. I
make no assertion of its truth (in fact, I don't think most people do
consciously minimize time, most of the time; they even "waste" it doing
things they "like" - see below). It is as if I had observed the relation
between temperature and pressure in a contained gas and began to speculate
about perfectly elastic tiny billiard balls whizzing around in perfectly
empty space, i.e., the kinetic theory of gases. In fact, the particles of
a gas tend to be not-perfectly-elastic, double-atom molecules, whizzing
around in not-perfectly-empty space. But it doesn't matter: the model is
good enough to make sense of the results (at the level of a simple gas
law). That is all I am doing with the assumption of
time-minimization.

I belabor this point because I think Sociology tends to misunderstand the
nature of scientific theory. Whereas most scientific explanation accounts
for patterns in the recurrent behavior of very large numbers of events
(atoms, molecules, cells, organisms, species, planets, galaxies),
Sociology tends toward a highly individual-voluntaristic view
of its "actors". When we think of crime, we usually picture a criminal
rather than a crime rate; when we think of adolescence or adolescents, we
seldom envision a population
pyramid.

I illustrate this tendency in class lectures with reference to an exhibit
of 12th century Chinese scroll paintings I saw at the DeYoung Museum in
San Francisco when I was an undergraduate. The right-hand side of Fig 8-4
is a badly beat-up poster I still have from that exhibit. The two pictures
on the left are details from the center-right (houses) and
lower-left (some people) of that poster. Some of paintings
were enormous, as much as 50 feet when fully unrolled. Ordinarily they
were scrolled, a slave standing on either side, slowly revealing portions,
so that the viewers, lounging before the changing scene, had a sense of
taking a journey through vast regions. The painting was supposed to
inspire thoughtful commentary, sometimes written on the painting itself in
the form of poetry.

FIG. 8-4. EMPHASIS ON THE EXTERNAL WORLD

I was fascinated by this view of the place of humans in nature. Here are
the sky, mountains, valleys, river gorges. There may be a forest or an
orchard, a bridge. Often people are shown amidst the buildings or natural
surroundings. Who are they? How are they grouped? The people in such a
picture are subject to all sorts of laws: meteorological, biological,
physical, nutritional, sociological. You don't have to enter their minds
to account for their behavior any more than you have to enter the minds of
molecules or monkeys to explain theirs. There is a structure in
which these people, and thousands of others elsewhere and earlier, live
out their lives.

The other rooms in the museum at that time were populated with European
works, mostly portraits, suggested here in Fig 8-5. Though some of them
had great artistic merit, their view of human nature was very different
from you get in Fig. 8-4 The entire emphasis here is on the individual and
his or her mood or character or claim-to-fame. About the only hint you get
that they even live in a world outside themselves might be in their title
(the 15th Earl implies that there were 14 before him) or the placement of
a pillar in the background (his pillar in his palace in
his world).

FIG. 8-5 EMPHASIS ON THE INDIVIDUAL

There is a similar contrast in point-of-view in much literature.
Moby Dick begins with the highly personal

Call me Ishmael. Some years agonever mind how long
preciselyhaving little or no money in my purse, and nothing particular
to interest me on shore, I thought I would sail about a little and see
the watery part of the world.

and ends after much struggle with

Now small fowls flew screaming over the yet
yawning gulf; a
sullen white surf beat against its steep sides; then all collapsed, and
the great shroud of the sea rolled on as it rolled five thousand years
ago.

Huckleberry
Finn, though less cosmic, moves similarly from the unique to the
recurrent. It begins with "You don't know about me...." and concludes:
"... Aunt Sally she's going to adopt me and civilize me, and I can't
stand it. I been there before."

I don't see how anyone could theorize about human behavior when
humans are viewed solely as willing, "motivated" individuals.
(Isn't "motivation" or "motive force" what got all the
pre-Galilean physicists back to Aristotle confused with regard to rest
versus motion?) Each individual is just that: a
genetically unique entity which, even if it were cloned a
thousand times, would still be unique in terms of subsequent
individual experience and personal "motivation". Predicting
behavior would be a matter of getting to know that individual
well enough to guess the next move correctly. The knowledge would
not generalize beyond that individual psyche. Your intense love
of football probably has little to say about mine for opera; your
need to golf has little to do with my need to solve puzzles. If
science is the study of recurrent events, then it has
nothing to say about individuals and their "likes" or "wills" or
"wants" or "motives". It should rather focus on the "common
denominator" which by assumption underlies all recurrent actions
and their patterns, something such as
time-minimization.

A theory of human behavior which "makes sense" at an individual level
(rational choice or libidinal repression) ought to be suspect. Theories
should account for patterns which appear in macro-level data. A
theory like time-minimization should not be judged in terms of intuitive
"reality"; it need only account for the empirical finding better than
any competitor.

Publishing the Derivation

I submitted the derivation to the American Sociological
Review on April 8, 1975. The rejection arrived on June 9. One
reviewer had no opinion regarding publication, but did offer some hints
for tightening the presentation. The other wrote that

The basic flaw in the analysis is the reliance on the
characteristics of an imaginary average region. A truly operational
model would use the real average distance to regional
centers...,

But theory is imaginary. When a theory is confirmed by direct
observation (e.g., cell theory through the microscope, the presence of
Neptune or Pluto with a telescope) it is no longer theory but fact. A
"truly operational model", as this reviewer seems to mean the words,
would be a description of an actual county. Is that even science? It
certainly isn't theory. The same reviewer didn't add to his credibility
when he pointed out that I hadn't "defined little d in the fraction
dT/dA" (the left side of Eq. 5 in the derivation above), and, even if I
had, it should be canceled from the fraction.[6]

After fruitless argument over these points with the ASR, I
submitted the paper to the American Journal of Sociology, January
16, 1976. It was rejected April 15. One referee described it as a
"useful fragment that ties together your previous work quite well" (you
could make the same remark about Newton's laws of motion). A second
found "work on area-density less than inspiring" (an interesting
criterion for judging scientific work). I asked for reconsideration and
received some discussion about the determinants of density (my
independent variable) and the statement

Who cares about county seats  more civilly, what's the
sociological payoff to a general theory with such limited
scope?

I mentioned the attention Durkheim had paid to precisely this topic (Chapter 4) and got nowhere, the final
rejection coming October 5, 1976.

A former undergraduate student, then studying for his doctorate in
Social Psychology at the University of Nevada, Phil Knowles, suggested I
submit the paper to Science which I did on November 1. There was
no hope of getting a positive review, but I knew I would get a quick and
competent one. Receipt was acknowledged on the 24th, and the review
arrived January 3, 1977. The referee wrote that it was "a very simple
and elegant derivation, which I think should be published." The editor,
however, said that "We receive each week about three times as many good
reports as we can publish...." and rejected it.

I wrote back on January 10 commenting only that, for a purportedly
interdisciplinary scientific journal, I saw very little social science
in Science. On January 18 I received the reply

We are glad to accept for publication as a report in
Science your paper on "Territorial Division: the Least-Time
Constraint Behind the Formation of Sub-National
Boundaries."

Nothing further by way explanation or commentary. I was ecstatic. It was
published the following April.[7] I have
received more, and more interesting, communication as a result of this
very brief publication in Science than I have from all my other
publications in Sociology journals combined. I have reproduced the article here, retaining as much of the original typography as possible (I couldn't create fully justified paragraphs however).

Earlier Derivations

Jay Callen has, over the years, brought to my attention three earlier
studies from other parts of the world which derive relations identical
or related to Eq. 1:

where v is the density of service centers and u is the
density of population. This is identical to an equation derived
differently in chapter 10 from Eq. 1.

* Earlier, in Greece, also using a least-distance model (which he
equated with least-effort) Virirakis[9]
produced

where P is the population and R is the radius of imaginary circular
areas in the i-th density category. Substituting DA = P (from the
definition for density) and A = πR2 (the area of a
circle), produces

which is virtually identical with Eq. 6.

* Earlier still, in Poland, Mycielski and Trzeciakowski,[10] in a derivation too long to be described
here, start from cost considerations and propose a method for locating
gasoline service stations efficiently (at least by their criteria). The
authors pass en route through two equations (on p. 68) compatible
with Eqs. 8 and 9.

Size-Density from Random Numbers

Charles GossmanLake Padden WA, March
2001

Around the time of the publication in Science, I developed, at the suggestion of my good friend and colleague Chuck Gossman (and initially to my horror), a derivation
of the -2/3 slope from random numbers.

Consider the size-density regression coefficient, the covariance of
log D and log A, divided by the variance of log D:

(10)

Re-writing the denominator and the numerator [11] enables us to "reduce" the
size-density slope, expressing it as a function of the variances of
log P and log A and the covariance of log P and log A:

(11)

Now draw three sets of random numbers. Label the first set P.
Multiply the second two sets (area is two dimensional) and label the
product set A. It follows that the covariance is

[6] As any first-quarter calculus student knows dy/dx is just
a symbolic way of saying the derivative of y with respect to x. it could
have been written y' or f'(x) rather than dy/dx. In particular, "little d" is not
a factor which can be "canceled". By the logic of my reviewer, the letter H
should be written as 1, since it looks like eleven divided by eleven. After
mentioning this review in a presentation to an audience of scientists, one
suggested that I should ask the reviewer to evaluate